Practical Electronics/Series RLC

A circuit of 3 components connected in series
 * [[Image:RLC series circuit.png|100px]]

Equilibrium Response
At Equilibrium, the sum of all voltages equal to zero
 * $$v_L + v_C + v_R = 0$$
 * $$L \frac{di}{dt} + \frac{1}{C} \int i dt + iR = 0$$
 * $$\frac{d^2i}{dt^2} + \frac{R}{L} \frac{di}{dt} + \frac{1}{LC}i = 0$$

The equation above can be written as below
 * $$\frac{d^2i}{dt^2} = - 2 \alpha \frac{di}{dt} - \beta i$$

With
 * $$\alpha = \frac{R}{2L}= \beta \gamma$$
 * $$\beta = \frac{1}{LC} = \frac{1}{T}$$
 * $$T=LC$$
 * $$\gamma=RC$$

Roots of 2nd ordered differential equation above


 * $$\alpha = \beta$$
 * $$ i(t) = Ae^{-\alpha t} $$


 * $$\alpha > \beta$$
 * $$ i(t) = Ae^{(-\alpha \pm \lambda)t}$$


 * $$\alpha < \beta$$
 * $$ i(t) = Ae^{(-\alpha \pm j \omega )t} =A(\alpha) \sin \omega t$$

Resonance Response
The total impedance of the circuit
 * $$Z = Z_R + Z_L + Z_C = R + 0 = R$$
 * $$i = \frac{V}{R}$$


 * $$Z_L = Z_C $$
 * $$j\omega L = \frac{1}{j\omega C} $$
 * $$\omega_o = \pm j \sqrt{\frac{1}{T}} $$
 * $$T=LC$$


 * At $$\omega_o = \pm j \sqrt{\frac{1}{T}} $$ the total impedance of the circuit is Z = R . Therefore, current is equal to    $$i = \frac{V}{R}$$
 * At $$\omega = 0 . Z_C = oo$$, Capacitor opens circuit . Therefore, current is equal to zero
 * At $$ \omega = oo . Z_L = oo$$, Inductor opens circuit . Therefore, current is equal to zero